1. Solving Equations with Variables on Both Sides
12-3
Solving Equations with Variables on Both Sides
Course 2
Warm Up
Problem of the Day
Lesson Presentation

2. Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Warm Up
Solve.
1. 6n + 8 – 4n = 20
2. –4w + 16 – 4w = –32
3. 25t – 17 – 13t = 67
4. 12 = 2(x + 7) + 4
n = 6
w = 6
t = 7
x = –3

3. Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Problem of the Day
You buy 1 cookie on the first day, 2 on the second day, 3 on the third day, and so on for 10 days. Your friend pays $10 for a cookie discount card and then buys 10 cookies at half price. You both pay the same total amount. What is the cost of one cookie?
$0.20

4. Learn to solve equations that have variables on both sides.
Course 2
12-3
Solving Equations with Variables on Both Sides
Learn to solve equations that have variables on both sides.

5. Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Additional Example 1: Using Inverse Operations to Group Terms with Variables
Group the terms with variables on one side of the equal sign, and simplify.
A. 60 – 4y = 8y
60 – 4y = 8y
60 – 4y + 4y = 8y + 4y
Add 4y to both sides.
60 = 12y
Simplify.
B. –5b + 72 = –2b
–5b + 72 = –2b
–5b + 5b + 72 = –2b + 5b
Add 5b to both sides.
72 = 3b
Simplify.

6. Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Check It Out: Example 1
Group the terms with variables on one side of the equal sign, and simplify.
A. 40 – 2y = 6y
40 – 2y = 6y
40 – 2y + 2y = 6y + 2y
Add 2y to both sides.
40 = 8y
Simplify.
B. –8b + 24 = –5b
–8b + 24 = –5b
–8b + 8b + 24 = –5b + 8b
Add 8b to both sides.
24 = 3b
Simplify.

7. Additional Example 2A: Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Additional Example 2A: Solving Equations with Variables on Both Sides
Solve.
7c = 2c + 55
7c = 2c + 55
7c – 2c = 2c – 2c + 55
Subtract 2c from both sides.
5c = 55
Simplify.
5c = 55 5
Divide both sides by 5.
c = 11

8. Additional Example 2B: Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Additional Example 2B: Solving Equations with Variables on Both Sides
Solve.
49 – 3m = 4m + 14
49 – 3m = 4m + 14
49 – 3m + 3m = 4m + 3m + 14
Add 3m to both sides.
49 = 7m + 14
Simplify.
49 – 14 = 7m + 14 – 14
Subtract 14 from
both sides.
35 = 7m
35 = 7m 7
Divide both sides by 7.
5 = m

9. Additional Example 2C: Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Additional Example 2C: Solving Equations with Variables on Both Sides 2 5
x = 1 x
– 12 2 5 1 5
x = x
– 12 2 5 x 15 – = 1
– 12 1 5
Subtract
x from both
sides. 1 5 x
–12 =
Simplify. 1 5
(5) x =
(5)(–12)
Multiply both sides by 5.
x = –60

10. Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Check It Out: Example 2A
Solve.
8f = 3f + 65
8f = 3f + 65
8f – 3f = 3f – 3f + 65
Subtract 3f from both sides.
5f = 65
Simplify.
5f = 65 5
Divide both sides by 5.
f = 13

11. Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Check It Out: Example 2B
Solve.
54 – 3q = 6q + 9
54– 3q = 6q + 9
54 – 3q + 3q = 6q + 3q + 9
Add 3q to both sides.
54 = 9q + 9
Simplify.
54 – 9 = 9q + 9 – 9
Subtract 9 from both sides.
45 = 9q
45 = 9q 9
Divide both sides by 9.
5 = q

12. Solving Equations with Variables on Both Sides
Course 2
12-3
Solving Equations with Variables on Both Sides
Check It Out: Example 2C
Solve. 2 3
w = 1 w
– 9 2 3 1 3
w = w
– 9 1 3 2 3 w 13 – = 1
– 9
Subtract
w from both
sides. 1 3 w –9 =
Simplify. 1 3
(3) w =
(3)(–9)
Multiply both sides by 3.
w = –27

13. Additional Example 3: Consumer Math Application
Course 2
12-3
Solving Equations with Variables on Both Sides
Additional Example 3: Consumer Math Application
Christine can buy a new snowboard for $ She will still need to rent boots for $8.50 a day. She can rent a snowboard and boots for $18.25 a day. How many days would Christine need to rent both the snowboard and the boots to pay as much as she would if she buys the snowboard and rents only the boots for the season?

14. Additional Example 3 Continued
Course 2
12-3
Solving Equations with Variables on Both Sides
Additional Example 3 Continued
Let d represent the number of days.
18.25d = d
Subtract 8.5d
from both sides.
18.25d – 8.5d = d – 8.5d
9.75d = 136.5
Simplify.
9.75d = 136.5
Divide both sides by 9.75.
9.75
9.75
d = 14
Christine would need to rent both the snowboard and the boots for 14 days to pay as much as she would have if she had bought the snowboard and rented only the boots.

15. Insert Lesson Title Here
Course 2
12-3
Solving Equations with Variables on Both Sides
Insert Lesson Title Here
Check It Out: Example 3
A local telephone company charges $40 per month for services plus a fee of $0.10 a minute for long distance calls. Another company charges $75.00 a month for unlimited service. How many minutes does it take for a person who subscribes to the first plan to pay as much as a person who subscribes to the unlimited plan?

16. Insert Lesson Title Here
Course 2
12-3
Solving Equations with Variables on Both Sides
Insert Lesson Title Here
Check It Out: Example 3 Continued
Let m represent the number of minutes.
75 = m
Subtract 40
from both sides.
75 – 40 = 40 – m
35 = 0.10m
Simplify. m
0.10 =
Divide both sides by 0.10.
350 = m
If you are going to use more than 350 minutes, it will be cheaper to subscribe to the unlimited plan.

17. Insert Lesson Title Here
Course 2
12-3
Solving Equations with Variables on Both Sides
Insert Lesson Title Here
Lesson Quiz: Part I
Group the terms with variables on one side of the equal sign, and simplify.
1. 14n = 11n + 81
2. –14k + 12 = –18k
Solve.
y = –4y – 19
4. –
3n = 81
4k = –12
y = –11 3 4 1 8
x =
x – 14
x = 16

18. Insert Lesson Title Here
Course 2
12-3
Solving Equations with Variables on Both Sides
Insert Lesson Title Here
Lesson Quiz Part II
5. Mary can purchase ice skates for $57 and then pay a $6 entry fee at the ice skating rink. She can also rent skates there for $3 and pay the entry fee. How many times must Mary skate to pay the same amount whether she purchases or rents the skates?
19 times